by Marek14 » Thu Dec 01, 2005 11:28 pm
To make things clearer: the original reason why I decided to include tigers and other beasts as toratopes was the behaviour of the toratope cuts.
When you cut a toratope with a coordinate hyperplane, you can get either a single toratope of lower dimension, or two toratopes of lower dimension, which are of the same kind. In this second case, the toratopes will differ EITHER in one coordinate of their centre, OR in one of their radii.
Let's look at an example: Sphere*circle has parameters
a(x,y,z),b(a,w)
We derive the cuts by simply eliminating one of the coordinates:
x-cut - a(y,z),b(a,w) - torus
y-cut - a(x,z),b(a,w) - torus
z-cut - a(z,y),b(a,w) - torus
w-cut - a(x,y,z),b(a) - two spheres.
Now, here's the rule for treating things like b(a): this means take a and replace it by two figures, displaced by +b and -b. In this case, the two spheres are concentric.
From this we can see that eliminating a coordinate will lead to SINGLE toratope if and only if the parameter containing that coordinate has at least two parameters left, and to two toratopes otherwise.
Let's ask a different question: What options do we have if we want a cut in shape of torus, or two torii?
A single torus looks like this: a(x,y),b(a,z). We can add the fourth coordinate to either a or b, leading to:
a(x,y,w),b(a,z)
or
a(x,y),b(a,z,w)
The first is sphere*circle, the second is circle*sphere. Both of these, therefore, can be cut to form a single torus.
However, what if I want to get two torii? I have five letters in the equations, and the torus can be duplicated through any of them. We only have four cases, since x and y are symmetrical, though. These all require another parameter, c:
c(x),a(c,y),b(a,z) are two torii displaced in the x dimension. It's a cut of c(x,w),a(c,y),b(a,z), or circle^3
a(x,y),c(a),b(a,z) are two torii with different external diameters. It's a cut of a(x,y),c(a,w),b(a,z), which is, again, circle^3.
a(x,y),b(a,z),c(b) are two torii with different internal diameters. It's a cut of a(x,y),b(a,z),c(b,w), once again - circle^3.
But there is one additional possibility, a(x,y),c(z),b(a,c)! This is two torii displaced in the z direction ("above" each other). And this is a cut of a(x,y),c(z,w),b(a,c), or tiger! This is the reason why tigers are neccessary to obtain the full theory - otherwise, some combinations of parameters would give nonexistent hyperplane cuts.
Observe the cuts in 5D:
A single GLOME - a(x,y,z,w) - is a cut of petaglome a(x,y,z,w,v)
Two displaced glomes - b(x),a(b,y,z,w) - are a cut of circle*glome b(x,v),a(b,y,z,w)
Two concentric glomes - a(x,y,z,w),b(a) - are a cut of glome*circle a(x,y,z,w),b(a,v)
A single SPHERE*CIRCLE - a(x,y,z),b(a,w) - is a cut of either a glome*circle a(x,y,z,v),b(a,w) or of a sphere^2 a(x,y,z),b(a,w,v)
Two sphere*circles displaced in x,y, or z coordinates - c(x),a(c,y,z),b(a,w) - are a cut of circle*sphere*circle c(x,v),a(c,y,z),b(a,w)
Two sphere*circles displaced in w coordinate - a(x,y,z),c(w),b(a,c) - are a cut of sphere tiger a(x,y,z),c(w,v),b(a,c)
Two concentric sphere*circles differing in diameter a - a(x,y,z),c(a),b(c,w) - are a cut of sphere*circle*circle a(x,y,z),c(a,v),b(c,w)
Two concentric sphere*circles differing in diameter b - a(x,y,z),b(a,w),c(b) - are a cut of sphere*circle*circle a(x,y,z),b(a,w),c(b,v)
A single CIRCLE^3 - a(x,y),b(a,z),c(b,w) - is a cut of either a sphere*circle*circle a(x,y,v),b(a,z),c(b,w), or of a circle*sphere*circle a(x,y),b(a,z,v),c(b,w), or of a circle*circle*sphere a(x,y),b(a,z),c(b,w,v).
Two circles^3 displaced in x or y dimension - d(x),a(d,y),b(a,z),c(b,w) - are a cut of circle^4 - d(x,v),a(d,y),b(a,z),c(b,w)
Two circles^3 displaced in z dimension - a(x,y),d(z),b(a,d),c(b,w) - are a cut of tiger*circle a(x,y),d(z,v),b(a,d),c(b,w)
Two circles^3 displaced in w dimension - a(x,y),b(a,z),d(w),c(b,d) - are a cut of torus tiger a(x,y),b(a,z),d(w,v),c(b,d)
Two concentric circles^3 differing in diameter a - a(x,y),d(a),b(d,z),c(b,w) - are a cut of circle^4 a(x,y),d(a,v),b(d,z),c(b,w)
Two concentric circles^3 differing in diameter b - a(x,y),b(a,z),d(b),c(d,w) - are a cut of circle^4 a(x,y),b(a,z),d(b,v),c(d,w)
Two concentric circles^3 differing in diameter c - a(x,y),b(a,z),c(b,w),d(c) - are a cut of circle^4 a(x,y),b(a,z),c(b,w),d(c,v)
A single TIGER - a(x,y),b(z,w),c(a,b) - is a cut of either a sphere tiger a(x,y,v),b(z,w),c(a,b), or of a circtiger a(x,y),b(z,w),c(a,b,v)
Two tigers displaced in any one dimension - d(x),a(d,y),b(z,w),c(a,b) - are a cut of a torus tiger d(x,v),a(d,y),b(z,w),c(a,b)
Two concentric tigers differing in diameter a or b - a(x,y),d(a),b(z,w),c(d,b) - are a cut of torus tiger a(x,y),d(a,v),b(z,w),c(d,b)
Two concentric tigers differing in diameter c - a(x,y),b(z,w),c(a,b),d(c) - are a cut of a tiger*circle a(x,y),b(z,w),c(a,b),d(c,v)
A single CIRCLE*SPHERE - a(x,y),b(a,z,w) - is a cut of either a sphere^2 a(x,y,v),b(a,z,w), or of a circle*glome a(x,y),b(a,z,w,v)
Two circle*spheres displaced in x or y dimension - c(x),a(c,y),b(a,z,w) - are a cut of circle*circle*sphere c(x,v),a(c,y),b(a,z,w)
Two circle*spheres displaced in z or w dimension - a(x,y),c(z),b(a,c,w) - are a cut of circtiger a(x,y),c(z,v),b(a,c,w)
Two concentric circle*spheres differing in diameter a - a(x,y),c(a),b(c,z,w) - are a cut of circle*circle*sphere a(x,y),c(a,v),b(c,z,w)
Two concentric circle*spheres differing in diameter b - a(x,y),b(a,z,w),c(b) - are a cut of circle*circle*sphere a(x,y),b(a,z,w),c(b,v)
You can see that thanks to inclusion of the beasts all possible cuts are real cuts of higher-dimensional toratopes.
Going in reverse, we find 1 cut for petaglome, 2 for glome*circle, 3 for sphere*circle*circle, 4 for circle^4, 2 for tiger*circle, 3 for circle*sphere*circle, 2 for sphere tiger, 3 for torus tiger, 2 for sphere^2, 3 for circle*circle*sphere, 2 for circtiger, and 2 for circle*glome. All 29 are accounted for.